The general aim of this project is to study the molecular mechanisms of free energy transduction in biology, such as in muscle contraction, kinesin movement, active transport, flagellar rotation, etc. During this period, our work was concentrated on the question of how a biological motor (such as myosin and kinesin) would carry out he repeated directional movements on a periodic biopolymer (such as myosin and kinesin) would carry out the repeated directional movements on a periodic biopolymer (such as actin and microtubule) by utilizing the free energy of ATP hydrolysis. These repeated directional movements are implicated in the theory of muscle contraction. Without going into complicated biological systems, we examined the simplest case: The chemically driven motility of a brownian particle. We showed with a simple model that an enzymatic brownian particle when coupled with a chemical reaction could move unidirectinally on a spatially periodic and asymmetric potential, if the intermediate states in the catalytic cyclic interacted with the potential differentially and that the chemical reaction was out of equilibrium. The general principle of the model could be tested experimentally and should provide useful in biomolecular separation. The model can also be generalized and used to study the motility of single biological motors in in vitro experiments, where brownian motion may play an important role.